Since the destabilisation of \(-e_3\)  if \(\beta >0\)  corresponds to a Hopf-instability of the \((\) purely essential\()\)  spectrum, it is effectively invisible in the coherent structure ODE, which detects changes in the linearization at zero temporal eigenvalue only. Visible from the PDE stability viewpoint is a transition of absolute spectrum through the origin in the complex plane of temporal eigenmodes, cf. [1]}. Now in the center case, the state \(-e_3\)  is already \(L^2\) -unstable since \(\alpha >0\)  as well as \(\mu <0\)  and \(h>\beta /\alpha \)  implies
\(h=h^\ast =\frac{\beta }{\alpha }-\frac{2\mu }{\alpha ^2}(1+\alpha ^2)>\frac{\beta }{\alpha }-\mu \) 
